Download - Pertemuan 15
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Pertemuan 15
Matakuliah : I0214 / Statistika MultivariatTahun : 2005Versi : V1 / R1
Analisis Ragam Peubah Ganda(MANOVA III)
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Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu :
• Mahasiswa dapat menerangkan konsep dasar analisis ragam peubah ganda (manova) C2
• Mahasiswa dapat menghitung manova satu klasifikasi C3
• Mahasiswa dapat melakukan uji Fisher dan uji Bartlette C3
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Outline Materi
• Konsep dasar analisis ragam peubah ganda (manova)
• Analisis ragam peubah ganda satu klasifikasi
• Uji Fisher
• Uji Bartlette
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<<ISI>>
Null Hypothesis
Univariate t-test:
H0 : 1 = 2 (population means are equal)
Multivariate case (2-group MANOVA):
H0 :
2p
22
12
1p
21
11
(population mean vectors are equal)
Main assumptions: normally distributed DVs, equal covariance
matrices across groups
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<<ISI>>
Test Statistic for 2-group MANOVA
Hotelling’s T2 : T2 = )yy()yy(nn
nn21
121
21
21
S
n1 : sample size in first group
n2 : sample size in second group
1y : vector of means of DVs in first group
2y : vector of means of DVs in second group
S : pooled within-group covariance matrix
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Hotelling’s T2 measures the between-group difference )yy( 21 , which
is weighted by the within-group covariance matrix S-1. The test works
as follows: From Hotellings T2, form
F = 2
21
21 Tp)2nn(
1pnn
F is the test statistic for testing whether there is a significant group
difference with respect to the whole vector y of dependent variables. F-
distributed with p and (n1 + n2 -p - 1) degress of freedom
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Tests of Significance
Wilks' Lambda
where Se represents the error SSCP matrix and Sh represents the
hypothesis SSCP matrix.
For Example In a fixed effects model, Sw is the Se for all effects.
While in the randoms effects model Sab is the Se for the main effects and Sw
for the interaction. If A is fixed and B is random th Sab is the Se for A main effect and Sw is the
Se for the B main effect and the interaction
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Rao's F Approximation
degrees of Freedom
Special Note Concerning s
If either the numerator or the deminator of s = 0 set s = 1
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Hotelling's Trace Criterion
Roy's Largest Latent Root
Pillai's Trace Criterion
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Which of these is "best?“ Schatzoff (1966)
Roy's largest-latent root was the most sensitive when population centroids differed along a single dimension, but was otherwise least sensative. Under most conditions it was a toss-up between Wilks' and Hotelling's criteria.
Olson (1976) Pillai's criteria was the most robust to violations of assumptions concerning homogeneity of the covariance matrix. Under diffuse noncentrality the ordering was Pillai, Wilks, Hotelling and Roy. Under concentrated noncentrality the ordering is Roy, Hotelling, Wilks and Pillai.
Final "Best" When sample sizes are very large the Wilks, Hotelling and Pillai become asymptotically equivalent
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<<ISI>>
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<<ISI>>
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<<ISI>>
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Tabel Manova
Sumber Variasi Matriks Jumlah Kuadrat dan Hasil Kali Silang
Derajat Bebas
Perlakuan 1
1
g
l ll
A n x x x x
1g
Residual
1 1
lng
lj l lj ll j
D x x x x
1
g
ll
n g
Total (terkoreksi)
1 1
lng
lj ljl j
A D x x x x
1
1g
ll
n
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Uji hipotesa
0 1 2: 0gH menyangkut generalized variance.
0H ditolak bila generalized variance
D
A D
kecil
( ditemukan oleh Wilks).
Distribusi yang eksak untuk diberikan dalam tabel
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Tabel Distribusi Wilks Lamda
Jumlah Variabel
Jumlah Grup
Distribusi sampling data multivariat
1p
2g
*
1, ( )*
1
1 l g
lg n
n gF
g
2p 2g
*
2( 1),2( 1)*
1 1
1 l
lg n g
n gF
g
1p 2g
1
*
,*
1 1l p
lp n
n pF
p
1p 3g
2
*
2 ,2*
2 1l p
lp n
n pF
p
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Bila 0H benar dan ln n besar:
*1 ln 1 ln2 2
Dp g p gn n
A D
berdistribusi mendekati Khi – kuadrat dengan derajat bebas 1p g .
Jadi, untuk ln besar, 0H ditolak pada tingkat signifikansi bila:
2( 1)
1 ln ( )2 p g
Dp gn
A D
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Jumlah Variabel
Jumlah Grup
Daerah penolakan H0
1p
2g
*
11, ( )*
1
1 n geg
n gF
g
2p 2g
1
*1
2( 1),2*
1 1
1 n glg
n gF
g
1p 2g
1
*1
* ,
1 1n pl
p
n pF
p
1p 3g
*
12 ,2 2*
2 1lp n p
n pF
p
Untuk ln besar.
0H ditolak dengan tingkat signifikansi bila
* 2( 1)1 ln ( )
2 p gp g
n
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<< CLOSING>>
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